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Identifier

St000568: Binary trees ⟶ ℤ

Values

[.,[.,.]] => 1

[[.,.],.] => 1

[.,[.,[.,.]]] => 1

[.,[[.,.],.]] => 2

[[.,.],[.,.]] => 1

[[.,[.,.]],.] => 2

[[[.,.],.],.] => 1

[.,[.,[.,[.,.]]]] => 1

[.,[.,[[.,.],.]]] => 2

[.,[[.,.],[.,.]]] => 2

[.,[[.,[.,.]],.]] => 2

[.,[[[.,.],.],.]] => 2

[[.,.],[.,[.,.]]] => 1

[[.,.],[[.,.],.]] => 2

[[.,[.,.]],[.,.]] => 2

[[[.,.],.],[.,.]] => 1

[[.,[.,[.,.]]],.] => 2

[[.,[[.,.],.]],.] => 2

[[[.,.],[.,.]],.] => 2

[[[.,[.,.]],.],.] => 2

[[[[.,.],.],.],.] => 1

[.,[.,[.,[.,[.,.]]]]] => 1

[.,[.,[.,[[.,.],.]]]] => 2

[.,[.,[[.,.],[.,.]]]] => 2

[.,[.,[[.,[.,.]],.]]] => 2

[.,[.,[[[.,.],.],.]]] => 2

[.,[[.,.],[.,[.,.]]]] => 2

[.,[[.,.],[[.,.],.]]] => 3

[.,[[.,[.,.]],[.,.]]] => 2

[.,[[[.,.],.],[.,.]]] => 2

[.,[[.,[.,[.,.]]],.]] => 2

[.,[[.,[[.,.],.]],.]] => 3

[.,[[[.,.],[.,.]],.]] => 2

[.,[[[.,[.,.]],.],.]] => 3

[.,[[[[.,.],.],.],.]] => 2

[[.,.],[.,[.,[.,.]]]] => 1

[[.,.],[.,[[.,.],.]]] => 2

[[.,.],[[.,.],[.,.]]] => 2

[[.,.],[[.,[.,.]],.]] => 2

[[.,.],[[[.,.],.],.]] => 2

[[.,[.,.]],[.,[.,.]]] => 2

[[.,[.,.]],[[.,.],.]] => 3

[[[.,.],.],[.,[.,.]]] => 1

[[[.,.],.],[[.,.],.]] => 2

[[.,[.,[.,.]]],[.,.]] => 2

[[.,[[.,.],.]],[.,.]] => 2

[[[.,.],[.,.]],[.,.]] => 2

[[[.,[.,.]],.],[.,.]] => 2

[[[[.,.],.],.],[.,.]] => 1

[[.,[.,[.,[.,.]]]],.] => 2

[[.,[.,[[.,.],.]]],.] => 3

[[.,[[.,.],[.,.]]],.] => 2

[[.,[[.,[.,.]],.]],.] => 3

[[.,[[[.,.],.],.]],.] => 2

[[[.,.],[.,[.,.]]],.] => 2

[[[.,.],[[.,.],.]],.] => 2

[[[.,[.,.]],[.,.]],.] => 3

[[[[.,.],.],[.,.]],.] => 2

[[[.,[.,[.,.]]],.],.] => 2

[[[.,[[.,.],.]],.],.] => 2

[[[[.,.],[.,.]],.],.] => 2

[[[[.,[.,.]],.],.],.] => 2

[[[[[.,.],.],.],.],.] => 1

[.,[.,[.,[.,[.,[.,.]]]]]] => 1

[.,[.,[.,[.,[[.,.],.]]]]] => 2

[.,[.,[.,[[.,.],[.,.]]]]] => 2

[.,[.,[.,[[.,[.,.]],.]]]] => 2

[.,[.,[.,[[[.,.],.],.]]]] => 2

[.,[.,[[.,.],[.,[.,.]]]]] => 2

[.,[.,[[.,.],[[.,.],.]]]] => 3

[.,[.,[[.,[.,.]],[.,.]]]] => 2

[.,[.,[[[.,.],.],[.,.]]]] => 2

[.,[.,[[.,[.,[.,.]]],.]]] => 2

[.,[.,[[.,[[.,.],.]],.]]] => 3

[.,[.,[[[.,.],[.,.]],.]]] => 2

[.,[.,[[[.,[.,.]],.],.]]] => 3

[.,[.,[[[[.,.],.],.],.]]] => 2

[.,[[.,.],[.,[.,[.,.]]]]] => 2

[.,[[.,.],[.,[[.,.],.]]]] => 3

[.,[[.,.],[[.,.],[.,.]]]] => 3

[.,[[.,.],[[.,[.,.]],.]]] => 3

[.,[[.,.],[[[.,.],.],.]]] => 3

[.,[[.,[.,.]],[.,[.,.]]]] => 2

[.,[[.,[.,.]],[[.,.],.]]] => 3

[.,[[[.,.],.],[.,[.,.]]]] => 2

[.,[[[.,.],.],[[.,.],.]]] => 3

[.,[[.,[.,[.,.]]],[.,.]]] => 2

[.,[[.,[[.,.],.]],[.,.]]] => 3

[.,[[[.,.],[.,.]],[.,.]]] => 2

[.,[[[.,[.,.]],.],[.,.]]] => 3

[.,[[[[.,.],.],.],[.,.]]] => 2

[.,[[.,[.,[.,[.,.]]]],.]] => 2

[.,[[.,[.,[[.,.],.]]],.]] => 3

[.,[[.,[[.,.],[.,.]]],.]] => 3

[.,[[.,[[.,[.,.]],.]],.]] => 3

[.,[[.,[[[.,.],.],.]],.]] => 3

[.,[[[.,.],[.,[.,.]]],.]] => 2

[.,[[[.,.],[[.,.],.]],.]] => 3

[.,[[[.,[.,.]],[.,.]],.]] => 3

[.,[[[[.,.],.],[.,.]],.]] => 2

[.,[[[.,[.,[.,.]]],.],.]] => 3

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$F_{2} = 2\ q$

$F_{3} = 3\ q + 2\ q^{2}$

$F_{4} = 4\ q + 10\ q^{2}$

$F_{5} = 5\ q + 30\ q^{2} + 7\ q^{3}$

$F_{6} = 6\ q + 70\ q^{2} + 56\ q^{3}$

$F_{7} = 7\ q + 140\ q^{2} + 252\ q^{3} + 30\ q^{4}$

Description

The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.

Code

def statistic(T):
    return T.hook_number()

Created

Aug 09, 2016 at 11:12 by Christian Stump

Updated

Aug 09, 2016 at 11:12 by Christian Stump